On the effect of rearrangement on complex interpolation for families of Banach spaces
aa r X i v : . [ m a t h . F A ] D ec ON THE EFFECT OF REARRANGEMENT ON COMPLEXINTERPOLATION FOR FAMILIES OF BANACH SPACES
YANQI QIUA
BSTRACT . We give a new proof to show that the complex interpola-tion for families of Banach spaces is not stable under rearrangement ofthe given family on the boundary, although, by a result due to Coifman,Cwikel, Rochberg, Sagher and Weiss, it is stable when the latter fam-ily takes only 2 values. The non-stability for families taking 3 valueswas first obtained by Cwikel and Janson. Our method links this prob-lem to the theory of matrix-valued Toeplitz operator and we are able tocharacterize all the transformations on T that are invariant for complexinterpolation at 0, they are precisely the origin-preserving inner func-tions. NTRODUCTION
This paper is a remark on the theory of complex interpolation for familiesof Banach spaces, developed by Coifman, Cwikel, Rochberg, Sagher andWeiss in [CCRSW82]. To avoid technical difficulties, we will concentrateon finite dimensional spaces.Let D = { z ∈ C : | z | < } be the unit disc with boundary T = ∂ D . Thenormalised Lebesgue measure on T is denoted by m . By an interpolationfamily, we mean a measurable family of complex N -dimensional normedspaces { E γ : γ ∈ T } , i.e., E γ is C N equipped with norm k · k γ and for each x ∈ C N , the function γ
7→ k x k γ defined on T is measurable. We shouldalso assume that R log + k x k γ dm ( γ ) < ∞ for any x ∈ C N . By definition,the interpolated space at 0 is E [0] := H ∞ ( T ; { E γ } ) /zH ∞ ( T ; { E γ } ) . That is, for all x ∈ C N , k x k E [0] = inf n ess sup γ ∈ T k f ( γ ) k E γ (cid:12)(cid:12)(cid:12) f : T → C N analytic , f (0) = x o . More generally, for any z ∈ D , the interpolated space at z for the family { E γ : γ ∈ T } is denoted by E [ z ] or { E γ : γ ∈ T } [ z ] whose norm is defined as follows. For any x ∈ C N , k x k E [ z ] := inf n ess sup γ ∈ T k f ( γ ) k E γ (cid:12)(cid:12)(cid:12) f : T → C N analytic , f ( z ) = x o . It is known (cf. [CCRSW82, Prop. 2.4]) that in the above definition, insteadof using ess sup γ ∈ T k f ( γ ) k E γ , we can use (cid:16) R k f ( γ ) k pE γ P z ( dγ ) (cid:17) /p for
Let ϕ : D → C be an inner function vanishing at 0. Itsboundary value is denoted again by ϕ : T → T . Then for any interpolationfamily { E γ : γ ∈ T } , the canonical identity: Id : { E γ : γ ∈ T } [0] → { E ϕ ( γ ) : γ ∈ T } [0] is isometric.Proof. The proof is routine, for details, see the last step in the proof ofTheorem 0.1 in the appendix. (cid:3)
Theorem 0.1 shows in particular that in the 2-valued case, the complexinterpolation is stable under rearrangement (the reader is referred to Lemma5.1 for the detail). We show that in the general case, this is not the case. Welearnt from the referee that this result was previously obtained by Cwikeland Janson in [ ? ] with a different method, the statement is at the bottom ofpage 214, the proof is from page 278 to page 283.Our method is simpler and it also yields a characterization of all the trans-formations on T that are invariant for complex interpolation at 0, they are EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 3 precisely the inner functions vanishing at 0. In other words, the converse ofProposition 0.2 holds.Here is how the paper is organised.In §
1, we recall a result from Helson and Lowdenslager’s papers [HL58,HL61] on the matrix-valued outer function F W : D → M N associated toa given matrix weight W : T → M N . This result allows us to give anapproximation formula for | F W (0) | when W is a small perturbation of theconstant weight I , where I is the identity matrix in M N .In §
2, we study the interpolation families consisting of distorted Hilbertspaces (i.e., C N equipped with norms k x k γ = k W ( γ ) / x k ℓ N for a.e. γ ∈ T ). We produce an explicit example of such a family for which complexinterpolation at 0 is not stable under rearrangement.Our main results are given in §
3, where we study some interpolation fami-lies consisting of 3 distorted Hilbert spaces. It is shown that in this restrictedcase, the complex interpolation at 0 is already non-stable under rearrange-ment. One advantage of our method is that we are able to characterize all thetransformations on T that are invariant for complex interpolation at 0, theyare precisely the inner functions Θ : T → T such that Θ(0) = b Θ(0) = 0 . § compatible Banach lattices. We also exhibit arather surprising non-stability example of interpolation family taking valuesin { X, X, X ∗ , X ∗ } .Finally, in the Appendix, we reformulate the argument of [CCRSW82]to prove Theorem 0.1, the proof somewhat explains why the 3-valued caseis different from the 2-valued case.1. A N APPROXIMATION FORMULA
In this section, we first recall some results from [HL58, §
5] and [HL61, § § §
12] in the forms that will be convenient for us, and then deducefrom them a useful formula.Let W : T → M N be a measurable positive semi-definite N × N -matrixvalued function such that tr ( W ) is integrable. Such a function should beconsidered as a matrix weight. Without mentioning, all matrix weights inthis paper satisfy: There exist c, C > such that cI ≤ W ( γ ) ≤ CI for a.e. γ ∈ T ; (1)where I is the identity matrix in M N . For such a matrix weight, let L W = L ( T , W ; S N ) be the set of functions f : T → M N for which k f k L W = Z tr (cid:16) f ( γ ) ∗ W ( γ ) f ( γ ) (cid:17) dm ( γ ) < ∞ . YANQI QIU
Clearly, L W is a Hilbert space.We will consider two subspaces H ( W ) ⊂ L W and H ( W ) ⊂ L W de-fined as follows: H ( W ) = { f ∈ L W | ˆ f ( n ) = 0 , ∀ n < } ,H ( W ) = { f ∈ L W | ˆ f ( n ) = 0 , ∀ n ≤ } . Given the assumption (1) on W , the identity map Id : L ( T ; S N ) → L W is an isomorphism, more precisely, c / k f k L ( T ; S N ) ≤ k f k L W ≤ C / k f k L ( T ; S N ) . (2)In particular, H ( T ; S N ) and H ( W ) are set theoretically identical but equippedwith equivalent norms.In the sequel, any element F ∈ H ( T ; S N ) will be identified with itsholomorphic extension on D , in particular, F (0) = b F (0) , the 0-th Fouriercoefficient.We recall the following theorem (a restricted form) of Helson and Low-denslager from [HL58] and [Hel64]. We denote by S N the spaces of N × N complex matrices equipped with the Hilbert-Schmidt norm. Theorem 1.1 (Helson-Lowdenslager) . Assume W a matrix weight satisfy-ing the assumption (1) . Then there exists F ∈ H ( T ; S N ) such that • F ( γ ) ∗ F ( γ ) = W ( γ ) for a.e. γ ∈ T . • F is a right outer function, that is, F · H ( T ; S N ) is dense in H ( T ; S N ) .Let Φ be the orthogonal projection of the constant function I to the subspace H ( W ) ⊖ H ( W ) ⊂ L W , i.e., Φ = P H ( W ) ⊖ H ( W ) ( I ) , then Φ( γ ) ∗ W ( γ )Φ( γ ) = | F (0) | for a.e. γ ∈ T . (3) Moreover, Φ and F and both invertible. If F and G are two (right) outer functions such that F ( γ ) ∗ F ( γ ) = G ( γ ) ∗ G ( γ ) = W ( γ ) for a.e. γ ∈ T , then there is a constant unitary matrix U ∈ U ( N ) such that F ( z ) = U G ( z ) for all z ∈ D . In particular, | F (0) | = | G (0) | is uniquely determined by W , as shown by the equation (3). Within all possible such outer functions,there is a unique one such that F (0) is positive, we will denote it by F W .Let Ψ = P H ( W ) ( I ) , where the orthogonal projection P H ( W ) is definedon the space L W . Clearly, we have Φ = I − Ψ . (4) EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 5
We have already known that set theoretically, H ( W ) = H ( T ; S N ) , andthey are equipped with equivalent norms, thus we have a Fourier series for Ψ ∈ H ( W ) = H ( T ; S N ) : Ψ = X n ≥ b Ψ( n ) γ n ; where the convergence is in H ( T ; S N ) and hence in H ( W ) .By definition, Ψ is characterized as follows. For any A ∈ M N and any n ≥ , we have h Ψ , γ n A i L W = h I, γ n A i L W , i.e., Z tr ( γ − n A ∗ W Ψ) dm ( γ ) = Z tr ( γ − n A ∗ W ) dm ( γ ) . Or equivalently, Z γ − n W Ψ dm ( γ ) = Z γ − n W dm ( γ ) , for n ≥ . (5)We denote by P + the orthogonal projection of L ( T ) onto the subspace H ( T ) . The generalized projection P + ⊗ I X on L p ( T ; X ) for < p < ∞ will still be denoted by P + . Note that P + is slightly different to theusual Riesz projection, the latter is defined as the orthogonal projectiononto H ( T ) . Similarly, we denote by P − the orthogonal projection onto H ( T ) and also its generalisation on L p ( T ; X ) when it is bounded. Withthis notation, the equation system (5) is equivalent to P + ( W Ψ) = P + ( W ) . (6) Key observation: If W is a perturbation of identity, that is, if there exists ameasurable function ∆ : T → M N such that ∆( γ ) ∗ = ∆( γ ) for a.e.γ ∈ T and k ∆ k L ∞ ( T ; M N ) < and W = I + ∆; then the equation (6) has the form Ψ + P + (∆Ψ) = P + (∆) . (7)The above equation can be solved using a Taylor series.To make the last sentence in the preceding observation rigorous, we in-troduce the following Toeplitz type operator: T ∆ : H ( T ; S N ) L ∆ −−→ L ( T ; S N ) P + −→ H ( T ; S N ); YANQI QIU where L ∆ : H ( T ; S N ) → L ( T ; S N ) is the left multiplication by ∆ on thesubspace H ( T ; S N ) . More precisely, ( L ∆ f )( γ ) = ∆( γ ) f ( γ ) for any f ∈ L ( T ; S N ) . Clearly, we have k T ∆ k ≤ k ∆ k L ∞ ( T ; M N ) < . The term P + (∆) in equation (7) should be treated as an element in H ( T ; S N ) ,then the equation (7) has the form ( Id + T ∆ )(Ψ) = P + (∆) . (8)Since k T ∆ k < , the operator Id + T ∆ is invertible. Thus equation (8) hasa unique solution Ψ ∈ H ( T ; S N ) = H ( W ) given by the formula: Ψ = ( Id + T ∆ ) − ( P + (∆)) = ∞ X n =0 ( − n T n ∆ ( P + (∆)); (9)where T ( P + (∆)) = P + (∆) , and the convergence is understood in thespace H ( T ; S N ) . Combining equations (3), (4) and (9), we deduce thefollowing formula: | F I +∆ (0) | = h I − ∞ X n =0 ( − n T n ∆ ( P + (∆)) i ∗ ( I + ∆) ×× h I − ∞ X n =0 ( − n T n ∆ ( P + (∆)) i . We summarize the above discussion in the following:
Proposition 1.2.
Let ∆ : T → M N be a measurable bounded selfadjointfunction such that k ∆ k L ∞ ( T ; M N ) < . Let ε ∈ [0 , , then we have | F I + ε ∆ (0) | = h I − ∞ X n =0 ( − n ε n +1 T n ∆ ( P + (∆)) i ∗ ( I + ε ∆) ×× h I − ∞ X n =0 ( − n ε n +1 T n ∆ ( P + (∆)) i . In particular, we have | F I + ε ∆ (0) | = I + ε b ∆(0) − ε X n ≥ | b ∆( n ) | + O ( ε ) , as ε → + . (10) EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 7
Proof.
It suffices to prove the approximation identity (10). We have | F I + ε ∆ (0) | = h I − εP + (∆) + ε T ∆ ( P + (∆)) + O ( ε ) i ∗ ( I + ε ∆) ×× h I − εP + (∆) + ε T ∆ ( P + (∆)) + O ( ε ) i = I + εR + ε R + O ( ε ) , as ε → + ; (11)where R = ∆ − P + (∆) − P + (∆) ∗ ,R = P + (∆) ∗ P + (∆) − ∆ P + (∆) − P + (∆) ∗ ∆+ T ∆ ( P + (∆))+ T ∆ ( P + (∆)) ∗ . For R , we note that since ∆ is selfadjoint, P − (∆) = P + (∆) ∗ and hence ∆ = P + (∆) + P + (∆) ∗ + b ∆(0) . (12)Thus R = b ∆(0) . For R , we note that since the left hand side of equation (11) is independentof γ ∈ T , the right hand side should also be independent of γ , hence R must be independent of γ , it follows that R = Z R ( γ ) dm ( γ )= Z (cid:16) P + (∆) ∗ P + (∆) − ∆ P + (∆) − P + (∆) ∗ ∆ (cid:17) dm ( γ )= − X n ≥ b ∆( n ) ∗ b ∆( n ) = − X n ≥ | b ∆( n ) | . (cid:3)
2. I
NTERPOLATION F AMILIES IN THE C ONTINUOUS C ASE
To any invertible matrix A ∈ GL N ( C ) is associated a Hilbertian norm k · k A on C N , which is defined as follows: k x k A = k Ax k ℓ N , for any x ∈ C N ; where ℓ N denotes the space C N with the usual Euclidean norm. Let usdenote ℓ A := ( C N , k · k A ) . We have the following elementary properties: • Let
A, B ∈ GL N ( C ) , then they define the same norm on C N if andonly if | A | = | B | . Thus, if U ∈ U ( N ) is a N × N unitary matrix,then k · k UA = k · k A . • We define a pairing ( x, y ) = P Nn =1 x n y n for any x, y ∈ C N , thenunder this pairing, we have the canonical isometries: ( ℓ A ) ∗ = ℓ A − T ; where A − T is the inverse of the tranpose matrix A T . YANQI QIU • We have the following canonical isometries: ℓ A = ℓ A and ℓ A ∗ = ℓ A ∗ ) − . Here we recall that, for a complex Banach space X , its complex conjugate X is defined to be the space consists of the same element of X , but withscalar multiplication λ · v = ¯ λv, for λ ∈ C , v ∈ X. Consider an N × N -matrix weight W . To such a weight is associated aninterpolation family { ℓ w ( γ ) : γ ∈ T } , where w ( γ ) = p W ( γ ) . The following elementary proposition will be used frequently:
Proposition 2.1.
For interpolation family { E γ : γ ∈ T } with E γ = ℓ w ( γ ) ,we have E [0] = ℓ F (0) , that is, k x k E [0] = k F (0) x k ℓ N , for all x ∈ C N ; where F ( z ) is any right outer function associated to the weight W .Proof. By the definition of right outer function associated to the weight W , F ( γ ) ∗ F ( γ ) = W ( γ ) for a.e. γ ∈ T . (13)For any x ∈ C N , define an analytic function f x : D → C N by f x ( z ) = F ( z ) − F (0) x, then f x (0) = x and for a.e. γ ∈ T , k f x ( γ ) k w ( γ ) = h W ( γ ) F ( γ ) − F (0) x, F ( γ ) − F (0) x i = h F ( γ ) ∗ F ( γ ) F ( γ ) − F (0) x, F ( γ ) − F (0) x i = k F (0) x k ℓ N . This shows that k f x k H ∞ ( T ; { E γ } ) ≤ k F (0) x k ℓ N , whence k x k E [0] ≤ k F (0) x k ℓ N = k x k ℓ F (0) . The converse inequality will be given by duality, it suffices to show that k x k E [0] ∗ ≤ k x k ℓ F (0) − T = k x k ( ℓ F (0) ) ∗ . Consider the dual interpolation family { E ∗ γ : γ ∈ T } = { ℓ w ( γ ) − T : γ ∈ T } , which is naturally given by the weight W ( γ ) − T = ( w ( γ ) − T ) ∗ w ( γ ) − T . By[CCRSW82, Th. 2.12], we have a canonical isometry { E ∗ γ : γ ∈ T } [0] = E [0] ∗ . EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 9
The identity (13) implies ( F ( γ ) − T ) ∗ F ( γ ) − T = W ( γ ) − T for a.e. γ ∈ T . Thus F ( z ) − T is the right outer function associated to the weight W ( γ ) − T .Then the same argument as above yields that k x k E [0] ∗ ≤ k x k ℓ F (0) − T = k x k ( ℓ F (0) ) ∗ . (cid:3) Remark 2.2.
More generally, assume that X is a (finite dimensional) normedspace such that M N ⊂ End ( X ) and k u · x k X = k x k X for any u ∈ U ( N ) . For instance X = S Np (1 ≤ p ≤ ∞ ) and M N acts on S Np bythe usual left multiplications of matrices. Consider the interpolation family E γ = ( X, k · k X ; A ( γ ) ) with k x k X ; A ( γ ) = k A ( γ ) · x k X for any γ ∈ T , then E [0] = ( X, k · k B (0) ) with k x k B (0) = k B (0) · x k X , where B ( z ) is any rightouter function associated to the matrix weight A ( γ ) ∗ A ( γ ) . The following result is probably known to the experts of prediction the-ory, since we do not find it in the literature, we include its proof.
Proposition 2.3.
The function { W ( γ ) : γ ∈ T } 7→ F W (0) or equivalently { W ( γ ) : γ ∈ T } 7→ | F (0) | is not stable under rearrangement. Moreprecisely, there exists a family { W ( γ ) : γ ∈ T } and a measure preservingmapping S : T → T , such that F W (0) = F W ◦ S (0) . Before we proceed to the proof of the proposition, let us mention that ifthe weight W ( γ ) takes only 2 distinct values, i.e., if W ( γ ) = A for γ ∈ Γ and W ( γ ) = A for γ ∈ Γ with T = Γ ∪ Γ a measurable partition, thena detailed computation shows that we have F W (0) = A / ( A − / A A − / ) m (Γ ) A / = A / ( A − / A A − / ) m (Γ ) A / . In particular, F W (0) = F W ◦ M (0) for any measure preserving mapping M : T → T . Of course, this can be viewed as a special case of Theorem 0.1. Thefact that we can calculate F W (0) efficiently in the above situation is due tothe fundamental fact that two quadratic forms can always be simultaneouslydiagonalized. Proof.
Fix r > , define two M -valued bounded analytic functions F , F : D → M by F ( z ) = (cid:20) (1 + r ) / r (1 + r ) − / z r ) − / (cid:21) ,F ( z ) = (cid:20) (1 + r ) − / r (1 + r ) − / z (1 + r ) / (cid:21) . Note that they are both outer since z → F ( z ) − and z → F ( z ) − arebounded on D . By a direct computation, F ( e iθ ) ∗ F ( e iθ ) = W ( e iθ ) = (cid:20) (1 + r ) / re iθ re − iθ (1 + r ) / (cid:21) ,F ( e iθ ) ∗ F ( e iθ ) = W ( e iθ ) = (cid:20) (1 + r ) / re − iθ re iθ (1 + r ) / (cid:21) . If we define S : T → T by S ( γ ) = γ , then S is measure preservingand W = W ◦ S . By noting that F (0) and F (0) are positive, we have F = F W and F = F W = F W ◦ S . However, F W ◦ S (0) = F (0) = F (0) = F W (0) . (cid:3) We denote W ( r ) ( e iθ ) := (cid:20) (1 + r ) / re iθ re − iθ (1 + r ) / (cid:21) , and let w ( r ) ( γ ) = p W ( r ) ( γ ) . The notation S : T → T will be reserved forthe complex conjugation mapping.An immediate consequence of Propositions 2.1 and 2.3 is the following: Corollary 2.4.
The interpolation family { e E ( r ) γ = ℓ w ( r ) ◦ S )( γ ) : γ ∈ T } isa rearrangement of the family { E ( r ) γ = ℓ w ( r ) ( γ ) : γ ∈ T } . The identitymapping Id : e E ( r ) [0] → E ( r ) [0] has norm k Id : e E ( r ) [0] → E ( r ) [0] k = (1 + r ) / . Proof.
Indeed, we have: k Id : e E ( r ) [0] → E ( r ) [0] k = sup x =0 k F W ( r ) (0) x k ℓ k F W ( r ) ◦ S (0) x k ℓ = k F W ( r ) (0) F W ( r ) ◦ S (0) − k M = (1 + r ) / . (cid:3) Remark 2.5.
By Corollary and a suitable discretization argument, wecan show that if J k = n e iθ : ( k − π ≤ θ < kπ o , for ≤ k ≤ , and let γ k ∈ J k be the center point on J k , then the interpolation families B ( r ) γ = ℓ w ( r ( γ k ) if γ ∈ J k and e B ( r ) γ = ℓ w ( r (¯ γ k ) if γ ∈ J k for r = p √ givedifferent interpolation space at 0, i.e., k Id : e B ( r ) [0] → B ( r ) [0] k > . We omit its proof, because in the next section, we give a better result byusing the formula obtained in § . EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 11
3. I
NTERPOLATION FOR THREE H ILBERT SPACES
In this section, we will show that complex interpolation is not stable evenfor a familiy taking only 3 distinct Hilbertian spaces. The starting pointof this section is Proposition 1. Our proof is somewhat abstract, but it ex-plains why the 3-valued case becomes different from the 2-valued case, theidea used in the proof will be applied further to get a characterization ofmeasurable transformations on T that perserve complex interpolation at 0. Theorem 3.1.
There are two different measurable partitions of the unit cir-cle: T = S ∪ S ∪ S = S ′ ∪ S ′ ∪ S ′ , m ( S k ) = m ( S ′ k ) , for k = 1 , , , and three constant selfadjoint matrices ∆ k ∈ M for k = 1 , , , such thatif we let ∆ = ∆ S + ∆ S + ∆ S and ∆ ′ = ∆ S ′ + ∆ S ′ + ∆ S ′ , then X n ≥ | b ∆( n ) | = X n ≥ | c ∆ ′ ( n ) | . Before turning to the proof of the above theorem, we state our main re-sult.
Corollary 3.2.
Let ∆ , ∆ ′ be as in Theorem . For < ε < k ∆ k ∞ , wedefine two matrix weights which are perturbation of identity: W ε = I + ε ∆ , W ′ ε = I + ε ∆ ′ . Denote w ε and w ′ ε the square root of W ε and W ′ ε respectively. Then thereexists ε < such that whenever < ε < ε , we have | F W ε (0) | = | F W ′ ε (0) | . Thus, whenever < ε < ε , the following two interpolation families { ℓ w ε ( γ ) : γ ∈ T } , { ℓ w ′ ε ( γ ) : γ ∈ T } have the same distribution and take only 3 distinct values. However, theinterpolation spaces at 0 given by these two families are different: ℓ F Wε (0) = ℓ F W ′ ε (0) . Proof.
This is an immediate corollary of Proposition 1.2 and Theorem 3.1.The last assertion follows from Proposition 2.1. (cid:3)
Remark 3.3.
We verify that in the two main cases where the interpolationis stable under rearrangement, the function ∆ P n ≥ | ˆ∆( n ) | is stableunder rearrangement. Note first that we have the following matrix identity: X n ≥ | b ∆( n ) | = Z | P + (∆) | dm. • ∆ is a 2-valued selfadjoint function, i.e, there isa measurable subset A ⊂ T and two selfadjoint matrices ∆ , ∆ ∈ M N , such that ∆ = ∆ A + ∆ A c then X n ≥ | b ∆( n ) | = Z | P + (∆) | dm = Z (cid:12)(cid:12)(cid:12) P + (cid:16) (∆ − ∆ )1 A + ∆ (cid:17)(cid:12)(cid:12)(cid:12) dm = | ∆ − ∆ | Z | P + (1 A ) | dm = m ( A ) − m ( A ) | ∆ − ∆ | , which depends on the measure of A but not the other structure of A .More generally, we note in passing that for any real valued f in L ( T ) the expression k P + ( f ) k = 2 P n ≥ | b f ( n ) | coincides withthe variance of f . • Rearrangement under inner functions: Let ϕ : T → T be theboundary value of an origin-preserving inner function. Assume ∆ : T → M N selfadjoint. Note that P + (∆ ◦ ϕ ) = P + (∆) ◦ ϕ and that ϕ preserves the measure m . Hence X n ≥ | \ (∆ ◦ ϕ )( n ) | = Z | P + (∆ ◦ ϕ ) | dm = Z | P + (∆) ◦ ϕ | dm = Z | P + (∆) | dm = X n ≥ | b ∆( n ) | . Proof of Theorem . Assume by contradiction that for any pair of 3-valuedselfadjoint functions ∆ and ∆ ′ as in the statement of Theorem 3.1, we have X n ≥ | b ∆( n ) | = X n ≥ | c ∆ ′ ( n ) | . (14)We make the following reduction. Step 1:
The above assumption implies that for any pair of functions, ∆ , ∆ ′ taking values in the same set of three matrices and having identical distri-bution, the equation (14) holds as well. Indeed, given such a pair, we can EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 13 consider the pair of selfadjoint functions which are still 3-valued: γ → (cid:20) γ ) ∗ ∆( γ ) 0 (cid:21) and γ → (cid:20) ′ ( γ ) ∗ ∆ ′ ( γ ) 0 (cid:21) . Then the square of the n -th Fourier coefficient becomes " | b ∆( n ) | | c ∆ ∗ ( n ) | and " | c ∆ ′ ( n ) | | c ∆ ′∗ ( n ) | respectively. The block (1, 1)-terms then give the desired equation. Step 2:
If we take N = 1 in the above step, then the conclusion is that forany pair of 3-valued scalar functions f, f ′ ∈ L ∞ ( T ) such that f d = f ′ , wehave P n ≥ | b f ( n ) | = P n ≥ | b f ′ ( n ) | , or equivalently, k P + ( f ) k = k P + ( f ′ ) k . Consequence I:
Under the above assumption, if ( A , A ) is a pair of twodisjoint measurable subsets of T , and ( A ′ , A ′ ) is another such pair such that m ( A ) = m ( A ′ ) and m ( A ) = m ( A ′ ) , then h P + (1 A ) , P + (1 A ) i L ( T ) = h P + (1 A ′ ) , P + (1 A ′ ) i L ( T ) (15)Indeed, if we define A := T \ ( A ∪ A ) and A ′ := T \ ( A ′ ∪ A ′ ) . For any α ∈ C , α = 0 , , consider f α = α A + 1 A + 0 × A , f ′ α = α A ′ + 1 A ′ + 0 × A ′ , then f α and f ′ α are two functions taking exactly 3 values 0, 1, α and f α d = f ′ α .Hence by the assumption, we have k αP + (1 A ) + P + (1 A ) k = k αP + (1 A ′ ) + P + (1 A ′ ) k , for any α ∈ C . (16)Note that for any measurable set A , since A is real, k P + (1 A ) k = m ( A ) − m ( A ) (17)Taking this in consideration, the equation (16) implies that ℜ (cid:16) α h P + (1 A ) , P + (1 A ) i (cid:17) = ℜ (cid:16) α h P + (1 A ′ ) , P + (1 A ′ ) i (cid:17) , for any α ∈ C , hence the equation (15) holds. Step 3:
We can deduce from our assumption the following consequence.
Consequence II:
For any pair of scalar functions f, f ′ ∈ L ∞ ( T ) (withoutthe assumption that they are both 3-valued), such that f d = f ′ , we have k P + ( f ) k = k P + ( f ′ ) k . Indeed, if f = n X k =1 f k A k , f ′ = n X k =1 f k A ′ k , where ( A k ) nk =1 are disjoint subsets of T , so is ( A ′ k ) nk =1 , moreover m ( A k ) = m ( A ′ k ) . By (15) and (17), we have k P + ( f ) k = n X k =1 | f k | · k P + (1 A k ) k + X ≤ k = l ≤ n f k f l h P + (1 A ) , P + (1 A ) i = n X k =1 | f k | · k P + (1 A ′ k ) k + X ≤ k = l ≤ n f k f l h P + (1 A ′ ) , P + (1 A ′ ) i = k P + ( f ′ ) k . Then by an approximation argument, more precisely, by using the fact thattwo functions f, f ′ ∈ L ( T ) such that f d = f can be approximated in L ( T ) by two sequences of simple functions ( g n ) and ( g ′ n ) such that g n d = g ′ n , wecan extend the above equality for pairs of equidistributed simple functionsto the general equidistributed pairs of functions, as stated in ConsequenceII. Step 4:
Now if we take f, f ′ ∈ L ∞ ( T ) to be f ( γ ) = γ and f ′ ( γ ) = γ , then f d = f ′ , but we have k P + ( f ) k = 1 = 0 = k P + ( f ′ ) k , which contradictsConsequence II. This completes the proof. (cid:3) Define T k := n e iθ | k − π ≤ θ < kπ o for k = 1 , , . We claim that in Theorem 3.1 and hence in Corollary 3.2, we can take forexample S = S ′ = T , S = S ′ = T , S = S ′ = T . Indeed, by the proof of Theorem 3.1, here we only need to show that h P + (1 T ) , P + (1 T ) i 6 = h P + (1 T ) , P + (1 T ) i . Since T ( γ ) = 1 T ( e − i π γ ) and T ( γ ) = 1 T ( e − i π γ ) , we have h P + (1 T ) , P + (1 T ) i = h P + (1 T ) , P + (1 T ) i . EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 15
Thus we only need to show that ℑ (cid:16) h P + (1 T ) , P + (1 T ) i (cid:17) = 0 . (18)Note that ℑ (cid:16) c T ( n ) c T ( n ) (cid:17) = sin π (1 − cos π )2 π n × , if n ≡ , if n ≡ − , if n ≡ . Hence ℑ (cid:16) h P + (1 T ) , P + (1 T ) i (cid:17) = 3 sin π (1 − cos π )2 π ∞ X k =0 k + 1(3 k + 1) (3 k + 2) , which is non-zero, as we expected.The same idea as in the proof of Theorem 3.1 yields the following charac-terization: combining with Proposition 0.2, we have characterized all mea-surable transformations on T that preserve complex interpolation at 0. Atthis stage, the proof is quite direct. Theorem 3.4.
Let
Θ : T → T be a measurable transformation. If for anyinterpolation family { E γ ; γ ∈ T } , we have { E γ : γ ∈ T } [0] = { E Θ( γ ) : γ ∈ T } [0] , then Θ is an inner function and ˆΘ(0) = 0 . Remark 3.5.
The main point of Theorem is to characterize all the trans-formations which preserve the interpolation spaces at origin.Proof.
It suffices to show that Θ ∈ H ∞ ( T ) , since by definition Θ( γ ) hasmodulus 1 for a.e. γ ∈ T . By Propositions 1.2, 2.1 and similar argumentsin the proof of Theorem 3.1, we have k P + ( f ◦ Θ) k = k P + ( f ) k , for any scalar function f ∈ L ∞ ( T ) . (19)Now take f ( γ ) = γ , we have k P + (Θ) k = k P + ( γ ) k = 0 , which impliesthat Θ ∈ H ∞ ( T ) and hence Θ ∈ H ∞ ( T ) . Then we can write Θ = b Θ(0) + P + (Θ) . In (19), if we take f ( γ ) = γ , then k P + (Θ) k = k P + ( γ ) k = 1 .Note that k Θ k = | b Θ(0) | + k P + (Θ) k , whence b Θ(0) = 0 . This completes the proof. (cid:3)
4. S
OME RELATED COMMENTS
Recall that an N -dimensional (complex) Banach space L is called a(complex) Banach lattice with respect to a fixed basis ( e , · · · , e N ) of L if it satisfies the lattice axiom: For any x k , y k ∈ C such that | x k | ≤ | y k | forall ≤ k ≤ N , k N X k =1 x k e k k L ≤ k N X k =1 y k e k k L . Thus in particular, k N X k =1 x k e k k L = k N X k =1 | x k | e k k L . The above fixed basis ( e , · · · , e N ) will be called a lattice-basis of L . Sucha Banach lattice L will be viewed as function spaces over the N -point set [ N ] = { , · · · , N } in such a way that e k corresponds to the Dirac functionat the point k . Thus for x, y ∈ L , we can write | x | ≤ | y | if | x k | ≤ | y k | forall ≤ k ≤ N , and log | x | = P Nk =1 log | x k | e k , suppose that x k = 0 for all ≤ k ≤ N .We will call { L γ = ( C N , k · k γ ) : γ ∈ T } a family of compatible Banach lattices, if there is an algebraic basis ( e , · · · , e N ) of C N whichis simultaneously a lattice-basis of L γ for a.e. γ ∈ T and such that < ess inf γ ∈ T k e k k γ ≤ ess sup γ ∈ T k e k k γ < ∞ for all ≤ k ≤ N. (20)In the sequel, the notation { L γ = ( C N , k · k γ ) : γ ∈ T } is reserved for afamily of compatible Banach lattices with respect to the canonical basis of C N .Complex interpolation at 0 for families of compatible Banach lattices isstable under any rearrangement. The proof of the following proposition isstandard. Proposition 4.1. If { L γ = ( C N , k · k γ ) : γ ∈ T } be an interpolation familyof compatible Banach lattices, then log k x k L [0] = inf Z log k f ( γ ) k γ dm ( γ ) , (21) where the infimum runs over the set of all measurable coordinate boundedfunctions f : T → C N , i.e., f k : T → C is bounded for all ≤ k ≤ N suchthat ( by convention log 0 := −∞ )log | x | ≤ Z log | f ( γ ) | dm ( γ ) . EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 17
In particular, if M : T → T is measure preserving and let { f L γ = L M ( γ ) : γ ∈ T } , then Id : L [0] → f L [0] is isometric.Proof. It suffices to show (21). Assume that x ∈ C N and k x k L [0] < λ .Without loss of generality, we can assume x k = 0 for all ≤ k ≤ N . Bythe definition of L [0] there exists an analytic function f = ( f , · · · , f N ) : D → C N such that f (0) = x and ess sup γ ∈ T k f ( γ ) k γ < λ. By (20), this implies in particular that f is coordinate bounded. Since z log | f k ( z ) | is subharmonic, we have log | x k | = log | f k (0) | ≤ Z log | f k ( γ ) | dm ( γ ) , for ≤ k ≤ N. Hence log | x | ≤ R log | f ( γ ) | dm ( γ ) . Obviously, R log k f ( γ ) k γ dm ( γ ) < log λ , whence inf Z log k f ( γ ) k γ dm ( γ ) ≤ log k x k L [0] . Conversely, assume that x ∈ C N and x k = 0 for all ≤ k ≤ N andlet f : D → C N be any coordinate bounded analytic function such that log | x | ≤ R log | f ( γ ) | dm ( γ ) . Then by (20), ess sup γ ∈ T k f ( γ ) k γ < ∞ andthere exists y ∈ C N such that | x | ≤ | y | and log | y | = Z log | f ( γ ) | dm ( γ ) . (22)Define u ( γ ) := log | f ( γ ) | . By assumption, x k = 0 and f k is bounded,hence log | f k | ∈ L ( T ) , so we can define the Hilbert transform of u k . Let ˜ u ( γ ) be the Hilbert transform of u ( γ ) and define g ( γ ) = e u ( γ )+ i ˜ u ( γ ) . Then g k ( γ ) = e u k ( γ )+ i ˜ u k ( γ ) is the boundary value of an outer function, hence log | g k (0) | = Z log | g k ( γ ) | dm ( t ) = Z u k ( γ ) dm ( γ ) = log | y k | . Thus | y | = | g (0) | . By [CCRSW82, Prop. 2.4], we have k y k L [0] = k g (0) k L [0] ≤ exp (cid:16) Z log k g ( γ ) k γ dm ( γ ) (cid:17) = exp (cid:16) Z log k f ( γ ) k γ dm ( t ) (cid:17) . It is easy to see that L [0] is a Banach lattice and by (22), k x k L [0] ≤ k y k L [0] . Thus log k x k L [0] ≤ log k y k L [0] ≤ Z log k f ( γ ) k γ dm ( γ ) . This proves the converse inequality. (cid:3)
Remark 4.2.
The preceding result should be compared with [CCRSW82,Cor. 5.2] , where it is shown that n L p γ ( X, Σ , µ ) : γ ∈ T o [ z ] = L p z ( X, Σ , µ ) , where /p z = R (1 /p γ ) P z ( dγ ) . Definition 4.3.
Let L = ( C N , k · k L ) be a symmetric Banach lattices, wedefine S L to be the space of N × N matrices equipped with the norm : k A k S L = k ( s ( A ) , · · · , s N ( A )) k L , where s ( A ) , · · · , S N ( A ) are singular numbers of the matrix A . If the Banach lattices L γ considered above are all symmetric, i.e., forany permutation σ ∈ S N and any x k ∈ C , k N X k =1 x k e σ ( k ) k L γ = k N X k =1 x k e k k L γ , then to each L γ is associated a Schatten type space S L γ = ( M N , k · k S L γ ) .The following proposition is classical (c.f. [Pie71]), we omit its proof. Proposition 4.4.
Let { L γ = ( C N , k · k γ ) : γ ∈ T } be an interpolationfamily of compatible symmetric Banach lattices and consider the associatedinterpolation family: { S L γ = ( M N , k · k S L γ ) : γ ∈ T } . Then for any z ∈ D , we have the following isometric identification Id : S L [ z ] → { S L γ } [ z ] . Combining Propositions 4.1 and 4.4, we have the following:
Corollary 4.5.
Consider the interpolation family { S L γ : γ ∈ T } . Let M : T → T be measure preserving and let { e S L γ = S L M ( γ ) : γ ∈ T } , then Id : { S L γ } [0] → { e S L γ } [0] is isometric. EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 19
The following proposition is related to our problem, see the discussionafter it.
Proposition 4.6.
Let { E γ : γ ∈ T } be an interpolation family of N -dimensional spaces such that there exist c, C > , for any x ∈ C N , c · min k | x k | ≤ k x k γ ≤ C · max k | x k | for a.e. γ ∈ T . Assume that Id : E ¯ γ → E γ ∗ is isometric for a.e. γ ∈ T . Then E [ ζ ] = ℓ N , for any ζ ∈ ( − , . Proof.
Fix ζ ∈ ( − , . For any x ∈ C N . Given any analytic function f : D → C N such that f ( ζ ) = x and k f k H ∞ ( { E γ } ) < ∞ . Since ζ = ¯ ζ , wehave f ( ζ ) = f ( ¯ ζ ) = x . The assumption on the interpolation family impliesthat the function z
7→ h f ( z ) , f (¯ z ) i is bounded analytic, hence log k x k ℓ N = log |h f ( ζ ) , f ( ¯ ζ ) i| ≤ Z log |h f ( γ ) , f (¯ γ ) i| P ζ ( dγ ) ≤ Z log (cid:16) k f ( γ ) k E γ k f (¯ γ ) k E ∗ γ (cid:17) P ζ ( dγ )= Z log (cid:16) k f ( γ ) k E γ k f (¯ γ ) k E ¯ γ (cid:17) P ζ ( dγ ) ≤ log (cid:16) k f k H ∞ ( { E γ } ) (cid:17) . Hence k x k ℓ N ≤ k f k H ∞ ( { E γ } ) . It follows that k x k ℓ N ≤ k x k E [ ζ ] . By duality, this inequality also holds in the dual case, hence we must have k x k ℓ N = k x k E [ ζ ] . (cid:3) Let Q j be the open arc of T in the j -th quadrant, i.e., Q j = n e iθ : ( k − π < θ < kπ o for ≤ j ≤ . Suppose that X and Y are N -dimensional, define two interpolation families { Z γ : γ ∈ T } and { e Z γ : γ ∈ T } by letting Z γ = X, γ ∈ Q Y, γ ∈ Q Y ∗ , γ ∈ Q X ∗ , γ ∈ Q , e Z γ = X, γ ∈ Q Y, γ ∈ Q X ∗ , γ ∈ Q Y ∗ , γ ∈ Q . By Proposition 4.6, Z [0] = ℓ N . For suitable choices of X and Y , we couldhave e Z [0] = ℓ N . More precisely, we have the following proposition. Proposition 4.7.
For any α ∈ T , define a × selfadjoint matrix δ α := (cid:20) αα (cid:21) . For < ε < , let w α,ε = ( I + εδ α ) / and X = ℓ w α,ε . Consider the weight W α,ε and the interpolation family generated by it as follows: W α,ε ( γ ) = I + εδ α , γ ∈ Q ( I + εδ α ) − , γ ∈ Q ( I + εδ α ) − , γ ∈ Q I + εδ α , γ ∈ Q ; g Z α,εγ = X, γ ∈ Q X ∗ , γ ∈ Q X ∗ , γ ∈ Q X, γ ∈ Q . There exists α ∈ T and < ε < , such that if < ε < ε then g Z α,ε [0] = ℓ N . Proof.
We have W α,ε ( γ ) = I + εδ α , γ ∈ Q I − εδ α + ε I + O ( ε ) , γ ∈ Q I − εδ α + ε I + O ( ε ) , γ ∈ Q I + εδ α , γ ∈ Q . Applying a slightly modified variant of the approximation equation (10),we have | F W α,ε (0) | = I + ε I − ε (cid:20) k P + ( h α ) k k P + ( h α ) k (cid:21) + O ( ε ); where h α = α Q − α Q − α Q + α Q . Assume by contradiction that g Z α,ε [0] = ℓ N for any α ∈ T and small ε .Then we must have k P + ( h α ) k = for any α ∈ T . In particular, α
7→ k P + ( h α ) k is a constant function on T . It follows that the following function is a constant function: C ( α ) = ℜh αP + (1 Q ) , − αP + (1 Q ) i + ℜh αP + (1 Q ) , αP + (1 Q ) i + ℜh− αP + (1 Q ) , − αP + (1 Q ) i + ℜh− αP + (1 Q ) , αP + (1 Q ) i . Clearly, by translation invariance of Haar measure, we have h P + (1 Q ) , P + (1 Q ) i = h P + (1 Q ) , P + (1 Q ) i = h P + (1 Q ) , P + (1 Q ) i , h P + (1 Q ) , P + (1 Q ) = h P + (1 Q ) , P + (1 Q ) i , hence C ( α ) = −ℜ n α (cid:16) h P + (1 Q ) , P + (1 Q ) i − h P + (1 Q ) , P + (1 Q ) i (cid:17)o . Then α C ( α ) is constant function if and only if h P + (1 Q ) , P + (1 Q ) i − h P + (1 Q ) , P + (1 Q ) i = 0 , EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 21 which is equivalent to ℑ (cid:16) h P + (1 Q ) , P + (1 Q ) i (cid:17) = 0 . (23)By a similar computation as in the proof of inequality (18), we have ℑ (cid:16) h P + (1 Q ) , P + (1 Q ) i (cid:17) = 4 π ∞ X k =0 k + 1(4 k + 1) (4 k + 3) , this contradicts (23), and hence completes the proof. (cid:3)
5. A
PPENDIX
Here we reformulate the argument of [CCRSW82] to emphasize the cru-cial role played by a certain inner function associated to the measurablepartition of the unit circle in proving Theorem 0.1. It follows from the pre-ceding that the analogous inner function for a measurable partition into 3subsets does not exist.
Lemma 5.1.
Suppose that Γ ∪ Γ is a measurable partition of T . Thenthere exists an inner function ϕ such that ϕ (0) = 0 . And ϕ (Γ ) ∪ ϕ (Γ ) isa partition of T into two disjoint arcs (up to negligible sets). Moreover, m ( ϕ (Γ )) = m (Γ ) and m ( ϕ (Γ )) = m (Γ ) . (24) Proof.
Since any origin-preserving inner function ϕ preserves the measure m on T (indeed note R T ϕ ( γ ) n dm ( γ ) = R T γ n dm ( γ ) ∀ n ∈ Z ), it suffices toshow the existence of an inner function satisfying the partition condition.Let v = 1 Γ : T → R be the characteristic function of Γ , its harmonicextension on D will also be denoted by v . Note that < v ( z ) < for any z ∈ D . Let ˜ v be the harmonic conjugate of v and define ψ = v + i ˜ v on D .Then ψ is an analytic map from D to S := { z ∈ C : 0 < ℜ ( z ) < } and hasnon-tangential limit ψ ( γ ) = v ( γ ) + i ˜ v ( γ ) , a.e. γ ∈ T . Thus ψ (Γ ) ⊂ ∂ and ψ (Γ ) ⊂ ∂ , where ∂ = { z ∈ C : ℜ ( z ) = 0 } and ∂ = { z ∈ C : ℜ ( z ) = 1 } . Let τ : S → D be a Riemann conformal mapping such that τ ( ψ (0)) = 0 . Notethat τ ( ∂ ) and τ ( ∂ ) are disjoint open arcs of T . Define ϕ = τ ◦ ψ : D → D . Then ϕ is an inner function such that ϕ (0) = 0 . We have ϕ (Γ ) ⊂ τ ( ∂ ) and ϕ (Γ ) ⊂ τ ( ∂ ) . Hence m ( ϕ (Γ )) ≤ m ( τ ( ∂ )) and m ( ϕ (Γ )) ≤ m ( τ ( ∂ )) . Since ϕ pre-serves the measure m , we have m ( ϕ (Γ )) + m ( ϕ (Γ )) ≤ m ( τ ( ∂ )) + m ( τ ( ∂ )) = 1 . Thus up to negligible sets, we have ϕ (Γ ) = τ ( ∂ ) and ϕ (Γ ) = τ ( ∂ ) . (cid:3) Proof of Theorem . Suppose Γ ∪ Γ is a measurable partition of the cir-cle and let the interpolation family { X γ : γ ∈ T } be such that X γ = Z for all γ ∈ Γ , X γ = Z for all γ ∈ Γ . By Lemma 5.1, we can find an inner function ϕ such that ϕ (0) = 0 and ϕ (Γ ) = J , ϕ (Γ ) = J up to negligible sets, where J ∪ J is a partitionof the circle into disjoint arcs. Consider the interpolation family of spaces { e X γ : γ ∈ T } such that e X γ = Z for all γ ∈ J , e X γ = Z for all γ ∈ J . Then by a conformal mapping, it is easy to see e X [0] = ( Z , Z ) θ , θ = m ( J ) = m (Γ ) . (25)We have e X ϕ ( γ ) = X γ for a.e. γ ∈ T . If x ∈ C N is such that k x k e X [0] < ,then by definition, there exists an analytic function f : T → C N such that f (0) = x and ess sup t ∈ T k f ( γ ) k e X γ < . Thusess sup γ ∈ T k ( f ◦ ϕ )( γ ) k X t = ess sup γ ∈ T k ( f ◦ ϕ )( γ ) k e X ϕ ( γ ) = ess sup γ ∈ T k f ( γ ) k e X γ < . Since ( f ◦ ϕ )(0) = f (0) = x , the above inequality shows that k x k X [0] < . By homogeneity, k x k X [0] ≤ k x k e X [0] . But if we consider the dual of theabove interpolation family, then we get the same inequality, hence we musthave k x k X [0] = k x k e X [0] . (26)By (26) and (25), we have X [0] = ( Z , Z ) θ , θ = m (Γ ) . (cid:3) By definition, a space is arcwise θ -Hilbertian if it can be obtained bycomplex interpolation of a family of spaces on the circle such that on anarc, the spaces are Hilbertian. Remark 5.2 (Communicated by Gilles Pisier) . The preceding argumentalso shows that, as conjectured in [Pis10] , of which we use the terminol-ogy, any θ -Hilbertian Banach space is automatically arcwise θ -Hilbertian,at least under suitable assumptions on the dual spaces, that are automatic EARRANGEMENT ON COMPLEX INTERPOLATION FOR FAMILIES 23 in the finite dimensional case. We merely indicate the argument in the lat-ter case. Consider a measurable partition Γ ∪ Γ of the unit circle with m (Γ ) = θ and a family of n -dimensional spaces { E γ | γ ∈ ∂D } suchthat E γ = ℓ n for any γ ∈ Γ but E γ is arbitrary for γ ∈ Γ . If ϕ is theinner function appearing in Lemma , and if we set F γ = E ϕ ( γ ) thenthe identity map Id : E [0] → F [0] is clearly contractive and F [0] is arc-wise θ -Hilbertian. Applying this to the dual family { E ∗ γ } in place of { E γ } and using the duality theorem from [CCRSW82, Th. 2.12 ] ) we find that Id : E [0] ∗ → F [0] ∗ is also contractive, and hence is isometric. This showsthat E [0] is arcwise θ -Hilbertian. A CKNOWLEDGEMENTS
The author is grateful to Gilles Pisier for stimulating discussions andvaluable suggestions, he would like to thank the referee for careful readingof the manuscript. The author was partially supported by the ANR grant2011-BS01-00801 and the A*MIDEX grant.R
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